Pin an object onto a character
Parenting to the character's vertex was what I was looking for. To do so:

1) Select everything.
2) Enter Edit mode (tab), and select the vertex you want to pin to.
3) Exit edit mode.
4) Select the object(s) you want to pin.
5) Shift-select the character.
6) Enter edit mode. The vertex should still be selected.
7) Press Ctrl-P to parent to vertex.

The objects will now follow around that vertex as it deforms.